Union, Intersection, Union, Intersection, Complement of an Event, Complement of an Event, OddsOdds

Dr .Hayk MelikyanDr .Hayk Melikyan

Department of Mathematics and CSDepartment of Mathematics and CS

melikyan@nccu.edumelikyan@nccu.edu

In this section, we will develop the rules of probability for compound events (more than one event) and will discuss probabilities involving the union of events as well as intersection of two events

The number of events in the union of A and B is equal to the The number of events in the union of A and B is equal to the number in A plus the number in B minus the number of events number in A plus the number in B minus the number of events

that are in both A and B.that are in both A and B.

Event B Event A

Sample space S N(A UB) = n(A) + n(B) – n(A B)

Addition RuleAddition Rule • If you divide both sides of the equation by n(S), the number

in the sample space, we can convert the equation to an equation of probabilities:

( ) ( ) ( ) ( )P A B P A P B P A B

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

n A B n A n B n A B

n S n S n S n S

Addition RuleAddition Rule

• A single card is drawn from a deck of cards. Find the probability that

• the card is a jack or club.• P(J or C) = p(J)+p(C)-

P(J and C)

Set of jacksSet of clubs

Jack and club (jack of clubs)

4 13 1 16 4

52 52 52 52 13

The events King and Queen are mutually exclusive. The events King and Queen are mutually exclusive. They cannot occur at the same time. So the They cannot occur at the same time. So the probability of a king probability of a king and queen and queen is zero. is zero.

• the card is king or queen

• =4/52+4/52 – 0= • 8/52= 2/13

Kings Queens

Kings and queens

( ) ( ) ( ) ( )P K Q p K P Q p K Q

Mutually exclusive events Mutually exclusive events

If A and B are mutually exclusive then

The intersection of A and B is the empty set

( ) ( ) ( )P A B p A p B

AB

Use a table to list outcomes of an experimentUse a table to list outcomes of an experiment• Three coins are tossed.  Assume they

are fair coins.  Give the sample space. Tossing three coins is the same experiment as tossing one coin three times. There are two outcomes on the first toss, two outcomes on the second toss and two outcomes on toss three. Use the multiplication principle to calculate the total number of outcomes: (2)(2)(2)=8 We can list the outcomes using a little “trick” In the far left hand column, write four H’s followed by four T’s. In the middle column, we write 2 H’s, then two T’s, two H’s , then 2 T’s. In the right column, write T,H,T,H,T,H,T,H . Each row of the table consists of a simple event of the sample space. The indicated row, for instance, illustrates the outcome {heads, heads, tails} in that order.

h h h

h h t

h t h

h t t

t h h

t h t

t t h

t t t

To find the probability of at least two tails, we mark To find the probability of at least two tails, we mark each row (outcome) that contains two tails or three each row (outcome) that contains two tails or three tails and divide the number of marked rows by 8 tails and divide the number of marked rows by 8 (number in the sample space) Since there are four (number in the sample space) Since there are four outcomes that have at least two tails, the probability outcomes that have at least two tails, the probability is 4/8 or ½ .is 4/8 or ½ .

h h h

h h t

h t h

h t t

t h h

t h t

t t h

t t t

Two dice are tossed. Two dice are tossed. What is the probability of a sum greater than 8 or doubles? What is the probability of a sum greater than 8 or doubles?

P(S>8 or doubles)=p(S>8) + p(doubles)-p(S>8 P(S>8 or doubles)=p(S>8) + p(doubles)-p(S>8 and and doubles) = doubles) = 10/36+6/36-2/36=14/36=7/18.10/36+6/36-2/36=14/36=7/18.

• (1,1), (1,2), (1,3), (1,4), (1,5) (1,6)• (2,1), (2,2), (2,3), (2,4), (2,5), (2,6)• (3,1), (3,2), (3,3), (3,4), (3,5), (3,6)• (4,1), (4,2), (4,3), (4,4), (4,5), (4,6)• (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)• (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

Circled elements belong to the intersection of the two events.

Complement RuleComplement Rule

Many times it is easier to compute the probability that A won’tMany times it is easier to compute the probability that A won’t

occur then the probability of event A.occur then the probability of event A. • Example: What is the probability

that when two dice are tossed, the number of points on each die will not be the same?

• This is the same as saying that doubles will not occur. Since the probability of doubles is 6/36 = 1/6, then the probability that doubles will not occur is 1 – 6/36 = 30/36 = 5/6.

( ) 1 ( )P not A P A

( ) ( ) 1P A p not A

OddsOdds • In certain situations, such as the gaming industry, it is

customary to speak of the odds in favor of an event E and the odds against E.

• Definition: Odds in favor of event

• Odds against E =

From adds to probability P(E) = a/(a + b)

• Example: Find the odds in favor of rolling a seven when two dice are tossed.

• Solution: The probability of a sum of seven is 6/36. So

'

( )

( )

P E

p E

'( )

( )

p E

P E

66 136

30 30 536

Problem 55Problem 55S = set of all lists of n birth months, n ≤ 12. Then n(S) = 12·12·...·12 (n times) = 12n.LetE="at least two people have the same birth Month”ThenE'= "no two people have the same birth month" n(E') = 12·11·10·...·[12 - (n - 1)]=